An up-against-the-ropes Jerry Yang and Yahoo! have rejected Microsoft’s unsolicited $31 a share offer and are expected to break the bad news to the software behemoth tomorrow, according to a source close to the situation.

The Yahoo! board, which met on Friday, feels the $44.6 billion bid “massively undervalues” the search titan, the sources added, and will spell that out in Monday’s letter.

A Yahoo! spokesman would not comment on word that its board rejected Microsoft’s offer, saying only the company is “pursuing all options.”

While the surprise Feb. 1 offer was a 62 percent premium over Yahoo!’s stock close the previous day, it is below Yahoo!’s 52-week high of $34.08. The Post reported last spring that Microsoft offered to buy Yahoo! for $36 a share but that Yang has rejected that overture.

The Yahoo! board is concerned, too, that the $31 offer doesn’t account for the risks the company would take by jumping into bed with Ballmer, as regulators might overturn the deal, this source said. While a Microsoft-Yahoo! combo would still trail industry leader Google in terms of the size and power of its search advertising business, the Micro-hoo duo would be the largest operator of email and instant messaging and could raise enough anti-trust concerns in Europe to scuttle the deal.

It is believed that Yang would still fight for a price higher than $36 a share. There has been speculation that Yahoo! would not accept anything less than $40 a share. That would be a premium of roughly 17 percent over Yahoo!’s 52-week high — but would cost Ballmer an extra $12 billion.

It is not known of Ballmer would give approval to upping the Microsoft by that much.

In addition to weighing a richer offer, Ballmer must also chew over the option of “going hostile” in an attempt to win over Yahoo! shareholder. While the hostile route — which includes taking the offer directly to each and every shareholder — has its downside, Ballmer is a bulldog and has indicated he is not afraid of a bare-knuckle brawl.

When we decide on a particular path to pursue, the CEO has said, “we keep on coming and coming and coming and coming.”

Going hostile could could anger Yahoo!’s brainiac engineer brigade and spark a brain drain. It could also bring Yahoo! to push regulators to put the kibosh on the deal.

Yang and his Yahoo! board have been scrambling the past nine days to find alternatives to the Microsoft bid.

The Post reported yesterday that Yang reached out to Time Warner after the media giant’s earnings call last week to gauge the company’s interest in exploring options between AOL and Yahoo!

Nothing came of the call.

One option for Yang is to outsource its search business to rival Google which could cut costs and improve revenue and profits. But that move would sent up a red flare to regulators over Google’s added muscle. It also makes Yahoo!’s position of being afraid to accept the offer on regulatory grounds a bit hallow.